Four colors are already forced at step 2. If all you want toshow is that at least 4 colors are needed, you can stop rightthere.

However if you continue past step 4 in the following way, you'll see you need a 5th color -- 4 colors is not enough.

5. 0=D is forced by 7=A, 8=B & 9=C

6. 1=E is forced by 8=B, 9=C, 0=D & 3=A

To explain the last step, note that vertex 1 is adjacent tovertices 8,9,0, so vertex 1 can't have one of the colors B,C,D.But vertex 1 is also adjacent to vertex 3, which was forcedearlier in the analysis (step 2) to have color A, hence vertex 1can't have color A either. Thus, vertex 1 can't have any of thecolors A,B,C,D.

It follows that the graph is not 4-colorable.

In any case, it's now _really_ unclear what you mean by a forced4-set. What's needed is a rigorous mathematical definition. Are you up to the task of providing such a definition using standardgraph-theoretic terminology?

Once the concept of forced 4-set is clarified to the point whereat least some of us understand what you're talking about, can you then state in a clear way what you're conjecturing withregard to 4-colorability or non-4-colorability?